12 The function \( f \) is defined for all real \( x \) by \( f(x)=\left\{\begin{array}{l}\frac{1}{x} \text { for } x>0 \\ x \mid \text { for } x \leq 0\end{array}\right. \) Find the set of values of \( x \) for which \( f(x) \leq 4 \)

To find the set of values of \( x \) for which \( f(x) \leq 4 \), we need to analyze the function \( f(x) \) defined as follows: \[ f(x) = \begin{cases} \frac{1}{x} & \text{for } x > 0 \\ x & \text{for } x \leq 0 \end{cases} \] We will consider both cases separately. ### Case 1: \( x > 0 \) In this case, \( f(x) = \frac{1}{x} \). We need to solve the inequality: \[ \frac{1}{x} \leq 4 \] To solve this, we can multiply both sides by \( x \) (noting that \( x > 0 \) means we do not change the direction of the inequality): \[ 1 \leq 4x \] Dividing both sides by 4 gives: \[ \frac{1}{4} \leq x \] Thus, for \( x > 0 \), the solution is: \[ x \in \left[\frac{1}{4}, \infty\right) \] ### Case 2: \( x \leq 0 \) In this case, \( f(x) = x \). We need to solve the inequality: \[ x \leq 4 \] Since \( x \) is already constrained to be less than or equal to 0, the solution for this case is: \[ x \in (-\infty, 0] \] ### Combining the Results Now we combine the results from both cases: 1. From Case 1: \( x \in \left[\frac{1}{4}, \infty\right) \) 2. From Case 2: \( x \in (-\infty, 0] \) Thus, the complete set of values of \( x \) for which \( f(x) \leq 4 \) is: \[ x \in (-\infty, 0] \cup \left[\frac{1}{4}, \infty\right) \] This is the final answer.